The question asks to find the no. of rearrangements of $ABCDE$ having exactly $1$ letter in its original position.
(Rearrangement of a set means any arrangement of the set including its original ordering).
Do we first have to fix an element (suppose $A$) and then have to find all the possible ways to arrange other $4$ elements in remaining $4$ positions?
Best Answer
We have to choose an element, then to choose a permutation of the remaining elements without fixed points (a derangement). Since there are $9$ permutations without fixed points in $S_4$ ($3$ elements with the cycle structure of $(1\,2)(3\,4)$ and $6$ elements with the cycle structure of $(1\,2\,3\,4)$), the answer is $5\cdot 9=45$.